light-cone (LC) variables

Transcription

1 light-cone (LC) variables 4-vector a µ scalar product metric LC basis : transverse metric 24-Apr-13 1

2 hadron target at rest inclusive DIS target absorbes momentum from γ * ; for example, if q z P z =0 P z = q M in DIS regime boost of 4-vector a µ a µ along z axis DIS regime direction + dominant direction - suppressed boost along z axis A = M hadron rest frame N.B. rapidity A = Q Infinite Momentum Frame (IFM) 24-Apr-13 2

3 A = M hadron rest frame A = Q Infinite Momentum Frame (IFM) LC kinematics boost to IFM definition : fraction of LC ( longitudinal ) momentum in QPM x ~ x B it turns out x N ~ x B + o(m/q) LC components not suppressed 24-Apr-13 3

4 rules at time x 0 =t=0 evolution in x 0 variables x Quantum Field Theory on the light-cone Rules at light-cone time x + =0 evolution in x + x -, x conjugated momenta k k +, k Hamiltonian k 0 k - field quantum.. Fock space.. 24-Apr-13 4

5 Dirac algebra on the light-cone usual representation of Dirac matrices so (anti-)particles have only upper (lower) components in Dirac spinor new representation in light-cone field theory definitions : projectors ok 24-Apr-13 5

6 project Dirac eq. does not contain time x + : χ depends from φ and A at fixed x + φ, A independent degrees of freedom good bad light-cone components component good independent and leading component bad dependent from interaction (quark-gluon) and therefore at higher order 24-Apr-13 6

7 quark polarization generator of spin rotations around z if momentum k z, it gives helicity γ 1, γ 2, γ 5 commute with P ± 2 possible choices : diagonalize γ 5 and Σ 3 helicity basis diagonalize γ 1 (or γ 2 ) transversity basis N.B. in helicity basis helicity = chirality for component good φ helicity = - chirality for component bad χ N.B. projector for transverse polarization we define 24-Apr-13 7

8 go back to OPE for inclusive DIS quark current 2MW µ X f e 2 f Z d 4 p ((p + q) 2 m 2 ) (p 0 + q 0 m) Tr (p, P ) µ (p/ + q/ + m) + (p, P ) (p/ + q/ m) µ ij im mn nj ij j i bilocal operator, contains twist 2 i j IFM (Q 2 ) isolate leading contribution in 1/Q equivalently calculate Φ on the Light-Cone (LC) 24-Apr-13 8

9 IFM: leading contribution 2MW µ = X f e 2 f Z d 4 p ((p + q) 2 m 2 ) (p 0 + q 0 m) N.B. p + ~ Q (p+q) ~ Q X Z Tr [ (p, P ) µ ( p + q + m) ] 1 2 e 2 f dp dp? Tr (p, P ) µ + p + =xp + f (analogously for antiquark) 24-Apr-13 9

10 (cont ed) decomposition of Dirac matrix Φ(p,P,S) on basis of Dirac structures with 4-(pseudo)vectors p,p,s compatible with Hermiticity and parity invariance Dirac basis 1I,i 5, µ, µ 5, µ,i µ 5 time-reversal 0 Tr (p, P ) µ + p + =xp + = 4g µ? (A 2 + xa 3 ) P + q f (x) similar for antiquark 24-Apr-13 10

11 (cont ed) x x B F 1 (x B ) QPM result Summary : W 1 response to transverse polarization of γ* bilocal operator Φ has twist 2 ; leading-twist contribution extracted in IFM selecting the dominant term in 1/Q (Q 2 ) ; equivalently, calculating Φ on the LC at leading twist (t=2) recover QPM result for unpolarized W µν ; but what is the general result for t=2? p + ~Q (p+q) - ~Q 24-Apr-13 11

12 Decomposition of Φ at leading twist Dirac basis ν Tr [γ + ] Tr [γ + γ 5 ] Tr [γ + γ i γ 5 ] 24-Apr-13 12

13 [ Trace of bilocal operator partonic density +] (x) = Z dp dp? Tr (p, P ) + p + =xp + = p 2 X n hn f (0) P i 2 (P + xp + P + n ) q f (x) LC good components probability density of annihilating in P> a quark with momentum xp + similarly for antiquark = probability of finding a (anti)quark with flavor f and fraction x of longitudinal (light-cone) momentum P + of hadron 24-Apr-13 13

14 in general : leading-twist projections (involve good components of φ ) twist 3 projections (involve good φ and bad χ components) Example: quark-gluon correlator suppressed not a density no probabilistic interpretation 24-Apr-13 14

15 probabilistic interpretation at leading twist helicity (chirality) projectors momentum distribution helicity distribution? 24-Apr-13 15

16 (cont ed) projector of transverse polarization (from helicity basis to transversity basis) more usual and comfortable notations: δq is net distribution of transverse polarization! unpolarized quark leading twist long. polarized quark transv. polarized quark 24-Apr-13 16

17 need for 3 Parton Distribution Functions al leading twist discontinuity in u channel of forward scattering amplitude parton-hadron target with helicity P emits parton with helicity p hard scattering parton with helicity p reabsorbed in hadron with helicity P A Pp,P p at leading twist only good components process is collinear modulo o(1/q) helicity conservation P+p = p+p 24-Apr-13 17

18 (cont ed) invariance for parity transformations A Pp,P p = A -P-p.-P -p invariance for time-reversal A Pp,P p = A P p,pp P p P p 1) ) ) constraints 3 A Pp,P p independent (+,+) (+,+) + (+,-) (+,-) f 1 (+,+) (+,+) - (+,-) (+,-) g 1 (+,+) (-,-) h 1 24-Apr-13 18

19 helicity basis transversity basis for good components ( twist 2) helicity = chirality hence h 1 does not conserve chirality (chiral odd) QCD conserves helicity at leading twist massless quark spinors λ = ± QCD conserves helicity at leading twist h 1 suppressed in inclusive DIS ± 24-Apr-13 19

20 different properties between f 1, g 1 and h 1 for inclusive DIS in QPM, correspondence between PDF s and structure fnct s but h 1 has no counterpart at structure function level, because for inclusive polarized DIS, in W A µν the contribution of G 2 is suppressed with respect to that of G 1 : it appears at twist 3 for several years h 1 has been ignored; common belief that transverse polarization would generate only twist-3 effects, confusing with g T in G 2 24-Apr-13 20

21 in reality, this bias is based on the misidentification of transverse spin of hadron (appearing at twist 3 in hadron tensor) and distribution of transverse polarization of partons in transversely polarized hadrons, that does not necessarily appear only at twist 3: Φ [Γ] long. pol. Φ [Γ] transv. pol. twist 2 γ + γ 5 g 1 i σ i+ γ 5 h 1 twist 3 i σ + - γ5 h L γ i γ 5 g T perfect crossed parallel between t=2 and t=3 for both helicity and transversity moreover, h 1 has same relevance of f 1 and g 1 at twist 2. In fact, on helicity basis f 1 and g 1 are diagonal whilst h 1 is not, but on transversity basis the situation is reversed: 24-Apr-13 21

22 h 1 is badly known because it is suppressed in inclusive DIS theoretically, we know its evolution equations up to NLO in α s there are model calculations, and lattice calculations of its first Mellin moment (= tensor charge). (Barone & Ratcliffe, Transverse Spin Physics, World Scientific (2003) ) only recently first extraction of parametrization of h 1 with two independent methods by combining data from semi-inclusive reactions: ( Anselmino et al., Phys. Rev. D (2007); hep-ph/ updated in arxiv: [hep-ph] Bacchetta, Courtoy, Radici, Phys. Rev. Lett (2011) JHEP 1303 (2013) 119 ) 1. h 1 has very different properties from g 1 2. need to define best strategies for extracting it from data 24-Apr-13 22

23 chiral-odd h 1 interesting properties with respect to other PDF g 1 and h 1 (and all PDF) are defined in IFM i.e. boost Q along z axis but boost and Galileo rotations commute in nonrelativistic frame g 1 = h 1 any difference is given by relativistic effects info on relativistic dynamics of quarks for gluons we define G(x) = momentum distribution ΔG(x) = helicity distribution but we have no transversity in hadron with spin ½ evolution of h 1 q decoupled from gluons! 24-Apr-13 23

24 (cont ed) axial charge tensor charge (not conserved) axial charge from C(harge)-even operator tensor charge from C-odd it does not take contributions from quark-antiquark pairs of Dirac sea summary: evolution of h 1q (x,q 2 ) is very different from other PDF because it does not mix with gluons evolution of non-singlet object moreover, tensor charge is non-singlet, C-odd and not conserved h 1 is best suited to study valence contribution to spin 24-Apr-13 24

25 relations between PDF s A Pp,P p [(+,+) (+,+)] + [(+,-) (+,-)] f 1 [(+,+) (+,+)] [(+,-) (+,-)] g 1 (+,+) (-,-) h 1 by definition f 1 g 1, h 1, f 1 0 (+,+) ± (-,-) 2 = A ++,++ + A - -,- - ± 2 ReA ++,-- 0 invariance for parity transformations A Pp,P p = A -P-p.-P -p A ++,++ = ½ (f 1 + g 1 ) A ++,-- = h 1 Soffer inequality valid for every x and Q 2 (at least up to NLO) 24-Apr-13 25

26 (cont ed) h 1 does not conserve chirality (chiral odd) h 1 can be determined by soft processes related to chiral symmetry breaking of QCD (role of nonperturbative QCD vacuum?) in helicity basis cross section must be chiral-even hence h 1 must be extracted in elementary process where it appears with a chiral-odd partner further constraint is to find this mechanism at leading twist how to extract transversity from data? 24-Apr-13 26

27 how to extract transversity from data? the most obvious choice: polarized Drell-Yan 24-Apr-13 27

28 Single-Spin Asymmetry (SSA) but = transverse spin distribution of antiquark in polarized proton antiquark from Dirac sea is suppressed and simulations suggest that Soffer inequality, for each Q 2, bounds A TT to very small numbers (~ 1%) better to consider but technology still to be developed (recent proposal PAX at GSI - Germany) otherwise. need to consider semi-inclusive reactions 24-Apr-13 28

29 alternative: semi-inclusive DIS (SIDIS) chiral-odd partner from fragmentation dominant diagram at leading twist ± chiral-odd transversiy ± in SIDIS {P,q,P h } not all collinear; convenient to choose frame where q T 0 sensitivity to transverse momenta of partons in hard vertex more rich structure of Φ Transverse Momentum Distributions (TMD) 24-Apr-13 29